From Fixed Bases to Structural Search: A Daopological Binary-Annealing Signal Analysis Paradigm and Its Formal Comparison with Fourier and Wavelet Transforms

The Fourier and wavelet transforms share an implicit premise: a signal should be projected onto a set of predefined analytic basis functions. This premise is highly successful for stationary or piecewise-stationary signals, but in strongly non-stationary, discontinuity-dominated, low-SNR regimes, the mismatch between the basis and the signal structure translates directly into representation error. This paper proposes a different signal-analysis paradigm: no analytic basis is predefined; instead, a signal segment is encoded into a binary discrete state (Yin–Yang), and the analysis is formulated as a multi-objective simulated-annealing search over a discrete state space, driven by a task energy function toward the signal’s structural attractor. The theoretical source is the axiomatic system of Daopology, which restates the Yijing as a binary state machine with algebraic-topological constraints rather than a divination text. We give formal definitions, a Hamiltonian construction, and operator semantics, and compare the paradigm with Fourier and wavelet transforms along four dimensions: representational prior, time–frequency resolution, discontinuity response, and computational complexity. The conclusion is complementarity, not replacement. A falsifiable experimental design is provided, but complete experimental data are withheld to preserve space for subsequent patent filings and empirical papers. Version 2.1 - Empirical Validation and Domain of Applicability UpdateThis version upgrades the original theoretical framework (v1.0) with rigorous empirical validations.Key updates include: Strict Blind Test: We removed all morphology priors (e.g., Hann window) and introduced a prior-free Gaussian blind decoder alongside a statistical dimensionality reduction encoder (MAD thresholding). Quantitative Results: At -5 dB SNR on asymmetric pulses (basis mismatch), the Daopological annealing engine outperforms the DWT (db4) baseline in both SNR gain (6.41 dB vs 6.28 dB) and pulse localization F1 score (0.75 vs 0.57). Honest Benchmarking: We explicitly report where DWT outperforms our method (e.g., pure square waves) and define the "Domain of Applicability". The paradigm is strictly advantageous for sparse, event-driven, highly non-stationary signals (like mechanical faults or ECG spikes), while remaining complementary to Fourier/Wavelet methods for continuous frequency-sweeping signals. Both English and Chinese LaTeX/PDF versions are updated.